The zero set of semi-invariants for extended Dynkin quivers

We show that the set of common zeros $\mathcal{Z}_{Q,\mathbf{d}}$ of all semi-invariants vanishing at 0 on the variety $\operatorname{rep}(Q,\mathbf{d})$ of all representations with dimension vector $\mathbf{d}$ of an extended Dynkin quiver $Q$ under the group $\operatorname{GL}(\mathbf{d})$ is a complete intersection if $\mathbf{d}$ is ``big enough''. In case $\operatorname{rep}(Q,\mathbf{d})$ does not contain an open $\operatorname{GL}(\mathbf{d})$-orbit, which is the case not considered so far, the number of irreducible components of $\mathcal{Z}_{Q,\mathbf{d}}$ grows with $\mathbf{d}$, except if $Q$ is an oriented cycle.